Calculus of Variations in Materials and Elasticity
Many models in continuum mechanics and materials science are formulated through the minimization of an energy. The relevant energies are often nonconvex, singularly perturbed, or dependent on several small scales. Consequently, minimizing sequences may oscillate, concentrate, form fine-scale patterns, or converge to an effective lower-dimensional or homogenized theory.
This course develops the mathematical tools used to analyze such problems. Topics include the direct method in the calculus of variations, convex duality, weak lower semicontinuity, quasiconvexity, relaxation, Young measures, Γ-convergence, homogenization, nonlinear elasticity, and the variational derivation of plate and shell models. Particular attention will be given to the geometric-rigidity approach of Friesecke, James, and Müller and to energy-driven pattern formation in phase-transforming and composite materials.
If time allowed, we will also study numerical approximation from a variational perspective: discrete compactness, convergence of discrete minimizers, numerical relaxation, and the distinction between convergence of energies, minimizers, stationary points, and algorithms.
This course develops the mathematical tools used to analyze such problems. Topics include the direct method in the calculus of variations, convex duality, weak lower semicontinuity, quasiconvexity, relaxation, Young measures, Γ-convergence, homogenization, nonlinear elasticity, and the variational derivation of plate and shell models. Particular attention will be given to the geometric-rigidity approach of Friesecke, James, and Müller and to energy-driven pattern formation in phase-transforming and composite materials.
If time allowed, we will also study numerical approximation from a variational perspective: discrete compactness, convergence of discrete minimizers, numerical relaxation, and the distinction between convergence of energies, minimizers, stationary points, and algorithms.
Lecturer
Date
14th September ~ 7th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 16:55 | Shuimo | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Basic knowledge about real analysis, functional analysis, and partial differential equations.
Syllabus
Tentative plan is as follows:
Week 1: Variational problems and the direct method
Week 2: Convexity and convex duality
Week 3: Nonlinear elasticity and nonconvexity
Week 4: Relaxation and generalized minimizers
Week 5: Fundamentals of Γ-convergence
Week 6: Geometric rigidity and dimension reduction
Week 7: Nonlinear bending theory for plates
Week 8: Hierarchy of plate and shell models
Week 9: Singular perturbations and pattern formation
Week 10: Homogenization and composite materials
Week 11: Energy-driven microstructure and scaling laws
Week 12: Variational convergence of numerical schemes
Week 1: Variational problems and the direct method
Week 2: Convexity and convex duality
Week 3: Nonlinear elasticity and nonconvexity
Week 4: Relaxation and generalized minimizers
Week 5: Fundamentals of Γ-convergence
Week 6: Geometric rigidity and dimension reduction
Week 7: Nonlinear bending theory for plates
Week 8: Hierarchy of plate and shell models
Week 9: Singular perturbations and pattern formation
Week 10: Homogenization and composite materials
Week 11: Energy-driven microstructure and scaling laws
Week 12: Variational convergence of numerical schemes
Reference
1. Bernard Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Springer.
2. Andrea Braides, Γ-Convergence for Beginners, Oxford University Press.
3. Philippe G. Ciarlet, Mathematical Elasticity Volume I, II, III
4. Andrea Braides and Anneliese Defranceschi, Homogenization of Multiple Integrals, Oxford University Press.
5. Georg Dolzmann, Variational Methods for Crystalline Microstructure—Analysis and Computation, Springer.
6. Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer.
7. Gero Friesecke, Richard D. James, and Stefan Müller, “A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity,” Communications on Pure and Applied Mathematics 55 (2002), 1461–1506.
8. Gero Friesecke, Richard D. James, and Stefan Müller, “A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence,” Archive for Rational Mechanics and Analysis 180 (2006), 183–236.
2. Andrea Braides, Γ-Convergence for Beginners, Oxford University Press.
3. Philippe G. Ciarlet, Mathematical Elasticity Volume I, II, III
4. Andrea Braides and Anneliese Defranceschi, Homogenization of Multiple Integrals, Oxford University Press.
5. Georg Dolzmann, Variational Methods for Crystalline Microstructure—Analysis and Computation, Springer.
6. Sören Bartels, Numerical Methods for Nonlinear Partial Differential Equations, Springer.
7. Gero Friesecke, Richard D. James, and Stefan Müller, “A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity,” Communications on Pure and Applied Mathematics 55 (2002), 1461–1506.
8. Gero Friesecke, Richard D. James, and Stefan Müller, “A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence,” Archive for Rational Mechanics and Analysis 180 (2006), 183–236.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
Chinese
, English